Article
| Title: | On the characterization of certain additive maps in prime $\ast $-rings (English) |
| Author: | Ashraf, Mohammad |
| Author: | Siddeeque, Mohammad Aslam |
| Author: | Shikeh, Abbas Hussain |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 2 |
| Year: | 2024 |
| Pages: | 549-565 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $\mathcal {A}$ be a noncommutative prime ring equipped with an involution `$*$', and let $\mathcal {Q}_{ms}(\mathcal {A})$ be the maximal symmetric ring of quotients of $\mathcal {A}$. Consider the additive maps $\mathcal {H}$ and $\mathcal {T} \colon \mathcal {A}\to \mathcal {Q}_{ms}(\mathcal {A})$. We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m+n)\mathcal {T}(a^2)=m\mathcal {T}(a)a^*+na\mathcal {T}(a)$ for all $a\in \mathcal {A}$ and $(m+n)\mathcal {H}(a^2)=m\mathcal {H}(a)a^*+na\mathcal {T}(a)$ for all $a\in \mathcal {A}$, then $\mathcal {H}=0$. (b) If $\mathcal {T}(aba)=a\mathcal {T}(b)a^*$ for all $a, b\in \mathcal {A}$, then $\mathcal {T}=0$. Furthermore, we characterize Jordan left $\tau $-centralizers in semiprime rings admitting an anti-automorphism $\tau $. As applications, we find the structure of generalized Jordan $*$-derivations in prime rings and generalize as well as improve all the results of A. Abbasi, C. Abdioglu, S. Ali, M. R. Mozumder (2022). (English) |
| Keyword: | prime ring |
| Keyword: | involution |
| Keyword: | generalized $(m, n)$-Jordan $*$-centralizer |
| MSC: | 16N60 |
| MSC: | 16W10 |
| MSC: | 47B47 |
| DOI: | 10.21136/CMJ.2024.0460-23 |
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| Date available: | 2024-07-10T14:56:19Z |
| Last updated: | 2024-07-15 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152457 |
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| Reference: | [1] Abbasi, A., Abdioglu, C., Ali, S., Mozumder, M. R.: A characterization of Jordan left *-centralizers via skew Lie and Jordan products.Bull. Iran. Math. Soc. 48 (2022), 2765-2778. Zbl 1517.16034, MR 4487734, 10.1007/s41980-021-00665-w |
| Reference: | [2] Beidar, K. I., III, W. S. Martindale: On functional identities in prime rings with involution.J. Algebra 203 (1998), 491-532. Zbl 0904.16012, MR 1622795, 10.1006/jabr.1997.7285 |
| Reference: | [3] Beidar, K. I., III, W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities.Pure and Applied Mathematics 196. Marcel Dekker, New York (1996). Zbl 0847.16001, MR 1368853 |
| Reference: | [4] Bennis, D., Dhara, B., Fahid, B.: More on the generalized $(m,n)$-Jordan derivations and centralizers on certain semiprime rings.Bull. Iran. Math. Soc. 47 (2021), 217-224. Zbl 1467.16021, MR 4215874, 10.1007/s41980-020-00377-7 |
| Reference: | [5] Brešar, M.: Functional identities and rings of quotients.Algebr. Represent. Theory 19 (2016), 1437-1450. Zbl 1361.16014, MR 3574001, 10.1007/s10468-016-9625-4 |
| Reference: | [6] Brešar, M., Chebotar, M. A., III, W. S. Martindale: Functional Identities.Frontiers in Mathematics. Birkhäuser, Basel (2007). Zbl 1132.16001, MR 2332350, 10.1007/978-3-7643-7796-0 |
| Reference: | [7] Fošner, A.: A note on generalized $(m,n)$-Jordan centralizers.Demonstr. Math. 46 (2013), 254-262. Zbl 1293.16033, MR 3089114, 10.1515/dema-2013-0456 |
| Reference: | [8] Herstein, I. N.: Jordan derivations of prime rings.Proc. Am. Math. Soc. 8 (1957), 1104-1110. Zbl 0216.07202, MR 0095864, 10.1090/S0002-9939-1957-0095864-2 |
| Reference: | [9] Herstein, I. N.: Topics in Ring Theory.University of Chicago Press, Chicago (1969). Zbl 0232.16001, MR 0271135 |
| Reference: | [10] Kosi-Ulbl, I., Vukman, J.: On $(m,n)$-Jordan centralizers of semiprime rings.Publ. Math. Debr. 89 (2016), 223-231. Zbl 1389.16079, MR 3529272, 10.5486/PMD.2016.7490 |
| Reference: | [11] Lanning, S.: The maximal symmetric ring of quotients.J. Algebra 179 (1996), 47-91. Zbl 0839.16020, MR 1367841, 10.1006/jabr.1996.0003 |
| Reference: | [12] Lee, T.-K., Lin, J.-H.: Jordan derivations of prime rings with characteristic two.Linear Algebra Appl. 462 (2014), 1-15. Zbl 1300.16044, MR 3255518, 10.1016/j.laa.2014.08.006 |
| Reference: | [13] Lee, T.-K., Lin, J.-H.: Jordan $\tau$-derivations of prime rings.Commun. Algebra 43 (2015), 5195-5204. Zbl 1327.16033, MR 3395699, 10.1080/00927872.2014.974103 |
| Reference: | [14] Lee, T.-K., Wong, T.-L.: Right centralizers of semiprime rings.Commun. Algebra 42 (2014), 2923-2927. Zbl 1293.16034, MR 3178052, 10.1080/00927872.2012.761711 |
| Reference: | [15] Lee, T.-K., Wong, T.-L., Zhou, Y.: The structure of Jordan *-derivations of prime rings.Linear Multilinear Algebra 63 (2015), 411-422. Zbl 1312.16046, MR 3273764, 10.1080/03081087.2013.869593 |
| Reference: | [16] Lee, T.-K., Zhou, Y.: Jordan *-derivations of prime rings.J. Algebra Appl. 13 (2014), Article ID 1350126, 9 pages. Zbl 1292.16037, MR 3153861, 10.1142/S0219498813501260 |
| Reference: | [17] Lin, J.-H.: Jordan $\tau$-derivations of prime GPI-rings.Taiwanese J. Math. 24 (2020), 1091-1105. Zbl 1467.16043, MR 4152657, 10.11650/tjm/191105 |
| Reference: | [18] Qi, X., Zhang, Y.: $k$-skew Lie products on prime rings with involution.Commun. Algebra 46 (2018), 1001-1010. Zbl 1441.16047, MR 3780213, 10.1080/00927872.2017.1335744 |
| Reference: | [19] Rowen, L.: Some results on the center of a ring with polynomial identity.Bull. Am. Math. Soc. 79 (1973), 219-223. Zbl 0252.16007, MR 0309996, 10.1090/S0002-9904-1973-13162-3 |
| Reference: | [20] Šemrl, P.: Quadratic functionals and Jordan *-derivations.Stud. Math. 97 (1991), 157-165. Zbl 0761.46047, MR 1100685, 10.4064/sm-97-3-157-165 |
| Reference: | [21] Šemrl, P.: Jordan *-derivations on standard operator algebras.Proc. Am. Math. Soc. 120 (1994), 515-518. Zbl 0816.47040, MR 1186136, 10.1090/S0002-9939-1994-1186136-6 |
| Reference: | [22] Siddeeque, M. A., Khan, N., Abdullah, A. A.: Weak Jordan *-derivations of prime rings.J. Algebra Appl. 22 (2023), Article ID 2350105, 34 pages. Zbl 07667259, MR 4556321, 10.1142/S0219498823501050 |
| Reference: | [23] Siddeeque, M. A., Shikeh, A. H.: On the characterization of generalized $(m,n)$-Jordan *-derivations in prime rings.Georgian Math. J. 31 (2024), 139-148. Zbl 07803155, MR 4698476, 10.1515/gmj-2023-2060 |
| Reference: | [24] Siddeeque, M. A., Shikeh, A. H.: A note on certain additive maps in prime rings with involution.(to appear) in Beitr. Algebra Geom. MR 4740676, 10.1007/s13366-023-00694-y |
| Reference: | [25] Vukman, J.: An identity related to centralizers in semiprime rings.Commentat. Math. Univ. Carol. 40 (1999), 447-456. Zbl 1014.16021, MR 1732490 |
| Reference: | [26] Vukman, J.: Centralizers on semiprime rings.Commentat. Math. Univ. Carol. 42 (2001), 237-245. Zbl 1057.16029, MR 1832143 |
| Reference: | [27] Vukman, J.: On $(m,n)$-Jordan centralizers in rings and algebras.Glas. Math., III. Ser. 45 (2010), 43-53. Zbl 1200.16051, MR 2646436, 10.3336/gm.45.1.04 |
| Reference: | [28] Vukman, J., Fošner, M.: A characterization of two-sided centralizers on prime rings.Taiwanese J. Math. 11 (2007), 1431-1441. Zbl 1148.16033, MR 2368661, 10.11650/twjm/1500404876 |
| Reference: | [29] Vukman, J., Kosi-Ulbl, I.: On centralizers of semiprime rings.Aequationes Math. 66 (2003), 277-283. Zbl 1073.16018, MR 2028564, 10.1007/s00010-003-2681-y |
| Reference: | [30] Zalar, B.: On centralizers of semiprime rings.Commentat. Math. Univ. Carol. 32 (1991), 609-614. Zbl 0746.16011, MR 1159807 |
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